Can cost increases increase competition? Asymmetric information and equilibrium prices.

* In this article, we present a model of competition under asymmetric information in which firms differ in their knowledge about the profitability of serving individual clients. We show that an increase in the production cost of the firm with superior information about customers can lead to increased competition and a reduction in market prices. This result extends the finding, common in the literature on asymmetric information, that the exit of a competitor who possesses superior private information may lead to an increase in competition. We show that the main results are robust to several extensions and obtain in a broad class of models. The key ingredients are that the market be characterized by information asymmetries among firms, and that these asymmetries represent the principal obstacle to competition.

We use our framework to examine some practical examples of industries where an increase in costs for some competitors has been associated with increased competition and lower prices. In addition, we note that the results in the model may also apply to other markets where information asymmetries are important. For example, there has been much literature to date on the ability of firms to poach each other's workers, and on the incentives firms may have to provide general training for their employees (see, e.g., Acemoglu and Pischke, 1998). Increasing the cost of production for one of the firms increases the liquidity of the labor market if the cost increase affects the firm employing more specialized labor whose talent is not publicly observable.

Finally, we note that the stylized nature of our model limits its suitability for welfare analysis. In particular, because all borrowers have identical reservation utility and unit demand for loans, any decrease in lending rates (including those due to increases in the informed firm's costs), while increasing borrowers' welfare, will have no impact on aggregate welfare. (16) In the extension in Section 5, an increase in the informed firm's costs will lead to a reduction in aggregate welfare because a larger mass of borrowers switches lender and therefore incurs the cost of switching. However, in models with negatively sloping aggregate demand for loans (for example, if borrowers have heterogeneous reservation utilities or a downward sloping individual demand for loans), the mechanism identified in this article could lead to increases in aggregate welfare associated with cost increases. Indeed, the increase in the volume of credit resulting from lower interest rates could more than compensate for the increase in the informed firm's cost.

Appendix

* Derivation of results for section on "private information for lender 2". As in the main text, the game is solved by backward induction. First, let [r.sub.i] be the interest rate charged by lender i to all unknown customers, and let [r.sub.i[delta]] be the interest rate charged to an old customer of type [theta]. Similarly to above, to retain a customer, lender i needs to offer it a rate no higher than that being offered by lender j, j [not equal to] i. Therefore, all old borrowers retained by lender i are charged a matching rate, [r.sub.i[theta]] = [r.sub.j]. At this rate, however, lender i will want to retain only borrowers of sufficiently high quality, for whom [r.sub.j] [theta] [greater than or equal to] 1 + [[delta].sub.i]. We can therefore define the threshold or cutoff quality level of old borrowers, [theta], needed to obtain financing from lender i as [[??].sub.i] [equivalent to] 1+[[delta].sub.i]/[r.sub.j], as long as lender j bids. If lender j does not make an offer, lender i can charge its old customers the maximum rate R without fear of losing them, so that the cutoff value [[??].sub.i] becomes 1+[[delta].sub.i]/R.

We now use this threshold 0 to define the pool of loan applicants to lender i: all new borrowers plus all old borrowers with repayment probability [theta] < [[??].sub.j] rejected by lender j. The analysis is analogous to that in the main text: lender i's profit on unknown borrowers, conditional on having the strictly lowest ("winning") rate, is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A2) Conditional on having the higher rate ("losing"), lender i, however, makes loans to lender j's rejected borrowers, so that its payoff is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A3)

Note that although [[pi].sub.i](r | l) (A3) is negative, [[pi].sub.i](r | w) (A1) can be positive for sufficiently large values ofr or low values of [[delta].sub.j].

We assume that the turnover parameter [lambda]. takes on a value in an interval ([[lambda].bar], [[bar.lambda]]), where 0 < [[lambda].bar] < [bar.[lambda]] < 1 (the boundaries [[lambda].bar] and [bar.[lambda]] are defined below). The proof of the following result follows from arguments similar to those in Proposition 1.

Proposition A l. For [lambda] [member of] ([[lambda].bar], [bar.[lambda]]), a unique equilibrium to the two-stage game exists and is characterized by a distribution function over strategies (interest rates and credit denial probability) for each lender, [F.sub.i](r), i = 1, 2, where [F.sub.i](r) = prob([r.sub.i] [less than or equal to] r). These distribution functions are continuous and strictly increasing on [[r.bar], R), where [r.bar] = (1 + [[delta].sub.2])+ 1/[square of ([lambda])] [square root of ([lambda][([[delta].sub.2] + 1).sup.2] + ([lambda] - 1)[[alpha].sub.1](1 + [[delta].sub.1]([[delta].sub.1] - 1 - 2[[delta].sub.2])], and are given by

[F.sub.1](r) = 1 + [[alpha].sub.1](1 - [lambda])(1 + [[delta].sub.1]) ([[delta].sub.1] - 2[[delta].sub.2]/2r[lambda](r[bar.[theta]] - (1 + [[delta].sub.2])) (A4)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A5)

The equilibrium has the following additional properties:

(i) Lender 2 makes zero expected profits. Lender 1 makes strictly positive expected profits from any new customers, as well as from its old customers.

(ii) Lender 2 refrains from bidding for new borrowers with positive probability, so that 1 - [F.sub.2](R) > 0 (the probability that lender 2 plays the strategy D is positive).

(iii) Lender 1 bids [r.sub.1] = R with positive probability.

Note now that for lender 1, with a larger information base,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A6)

Therefore, [partial derivative]E[[r.sub.1]]/[partial derivative][[delta].sub.1] < 0.

For lender 2, start by noting that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A7)

In other words, it is the same expression as for the case when lender 2 has no market share, plus an additional term that reflects lender 2's market share. We now have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A8)

We already established that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. For the rest, note that, because [[delta].sub.1] [greater than or equal to] [[delta].sub.2], the term [[delta].sub.2] - 1 - 2[[delta].sub.1] will always be negative. The term 1/[r.bar] - 1/r, however, must be positive. Using the definition of [r.bar] = (1 + [[delta].sub.2]) + 1/[square root of ([lambda])] [square root of ([lambda][([[delta].sub.2] + 1).sup.2] + ([lambda] - 1)[[alpha].sub.1](1 + [[delta].sub.1]([[delta].sub.1] - 1 - 2[[delta].sub.2)], we see that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A9)

Putting these things together, we can conclude that the entire expression for [partial derivative][F.sub.2]/[partial derivative][[delta].sub.1] will be positive for (i) [[alpha].sub.2] sufficiently small; and (ii) [[delta].sub.2] not too small relative to [[delta].sub.1]. This is summarized in the following result.

Proposition A2. For fixed [[delta].sub.1] [greater than or equal to] [[delta].sub.2] [greater than or equal to] 1, there exists an [[??].sub.2] > 0 such that for as [[alpha].sub.2], < [[??].sub.2], [partial derivative][F.sub.2]/[partial derivative][[delta].sub.1] > 0.

Coupled with the finding that [partial derivative][r.bar]/[partial derivative][[delta].sub.1] [less than or equal to] 0, this result allows us to conclude that, for [[alpha].sub.2] < [[??].sub.2], [partial derivative]E[[r.sub.2]]/[partial derivative][[delta].sub.1] < 0, as desired.

To calculate the bounds [[lambda].bar] and [bar.[lambda]], start with the lower bound [[lambda].bar], the value for which information asymmetries are so large that lender 2 cannot make positive profits. In other words, we need the value of [lambda] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A10)

This can be solved to yield

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A11)

To get the upper bound [bar.[lambda]], we need to find the value of [lambda] for which lender 2 has a sufficiently large cost advantage that it squeezes lender 1 out of the market entirely. Start by finding the rate offered by lender 2 such that even if it wins, lender 1 just breaks even. In other words,

[[pi].sub.1](r|w) = [lambda](r[bar.[theta]] - (1 + [[delta].sub.1])) + (1 - [lambda]) [[alpha].sub.2] 1 + [[delta].sub.2/r 1/2 ([[delta].sub.2] 1 - 2[[delta].sub.1]) = 0. (A12)

Solving for this rate and using the fact that [bar.[theta]] = 1/2, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A13)

We can now solve for the value of [delta] such that lender 2 would just make positive profits by offering that rate,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A14)

Solving for [lambda] gives us the upper bound [bar.[lambda]].